We consider sender–receiver games in which the sender has finitely many types and the receiver makes a decision in a compact set. The new feature is that, after the cheap talk phase, the receiver makes a proposal to the sender, which the latter can reject in favor of an outside option. We focus on situations in which the sender’s approval is absolutely crucial to the receiver, namely, on equilibria in which the sender does not exit at the approval stage. A nonrevealing equilibrium without exit may not exist. Our main results are that if the sender has only two types or if the receiver’s preferences over decisions do not depend on the type of the sender, there exists a (perfect Bayesian Nash) partitional equilibrium without exit, in which the sender transmits information by means of a pure strategy. The previous existence results do not extend: we construct a counter-example (with three types for the sender and type-dependent utility functions) in which there is no equilibrium without exit, even if the sender can randomize over messages. We establish additional existence results for (possibly mediated) equilibria without exit in the three type case.

我们考虑发送方－接收方游戏，其中发送方具有有限的多种类型，并且接收方在紧凑的集合中做出决策。新功能是，在廉价通话阶段之后，接收方向发送方提出建议，发送方可以拒绝该发送方以使用外部选项。我们关注发送者的批准对接收者绝对至关重要的情况，即发送者在批准阶段不退出的均衡。没有退出的非公开均衡可能不存在。我们的主要结果是，如果发送方只有两种类型，或者如果接收方对决策的偏好不取决于发送方的类型，则存在（完全贝叶斯纳什）分区均衡而没有退出，发送方通过这种方式传输信息纯策略。先前的存在结果不会扩展：我们构造了一个反例（具有三种类型的发送方和依赖于类型的效用函数），即使发送方可以根据消息进行随机化，也没有退出就不会达到平衡。我们建立了（可能是介导的）均衡的其他存在结果，而在三种情况下都没有退出。